My key was password protected which was causing the problem, a working file is now listed below (for help of future googlers)
FROM ubuntu
MAINTAINER Luke Crooks "[email protected]"
# Update aptitude with new repo
RUN apt-get update
# Install software
RUN apt-get install -y git
# Make ssh dir
RUN mkdir /root/.ssh/
# Copy over private key, and set permissions
# Warning! Anyone who gets their hands on this image will be able
# to retrieve this private key file from the corresponding image layer
ADD id_rsa /root/.ssh/id_rsa
# Create known_hosts
RUN touch /root/.ssh/known_hosts
# Add bitbuckets key
RUN ssh-keyscan bitbucket.org >> /root/.ssh/known_hosts
# Clone the conf files into the docker container
RUN git clone [email protected]:User/repo.git